Does the cycle of Adam repeat throughout human history, with a prophet like Muhammad (saw) in every age?

No. Humanity as a whole develops upwards, and the Holy Prophet Muhammad (saw) stands at the peak of this development. Someone like him could not have appeared in any earlier age.

The history of humanity has left traces behind, and these traces can be read. They show a steady development of awareness, behaviour, discipline and character. There have been many setbacks, and some periods fell far behind. Yet on the whole the path leads upwards.

That is why no earlier period of history equals this one. Every earlier time was still lagging behind in this development. Whoever stands at the peak of such a journey cannot appear at its beginning. So someone like the Holy Prophet Muhammad (saw) could not have come in any earlier age. The first verse above calls him the Seal of the Prophets.

Religion has a great aim in all this. Humanity began as one community believing in the One God, as the second verse says. Then it divided in many directions. In the end it is meant to come together again in one religion. This will not happen easily, but only through great effort and much sacrifice. In this way a circle is completed.

What comes after that rests with God. If humanity receives this gift and then rejects it once more, no further prophet will come to rescue it. God can then bring forth a new creation, as the third verse says. What that would be like, we do not know. What matters for us is the path that lies before us now.